**EC Cryptography Tutorials - Herong's Tutorial Examples** - Version 1.00, by Dr. Herong Yang

Reduced Elliptic Curve Group - E127(-1,3)

This section provides an example of a reduced Elliptic Curve group E127(-1,3). An example of addition operation is also provided.

Let's take a look at another reduced elliptic curve group, E_{127}(-1,3),
as discussed in "Elliptic Curve Cryptography: finite fields and discrete logarithms"
by Andrea Corbellini at
andrea.corbellini.name/2015/05/23/elliptic-curve-cryptography-finite-fields-and-discrete-logarithms/.

Here is the reduced elliptic curve group using modular arithmetic of prime number 127,
E_{127}(-1,3):

y^{2}= x^{3}- x + 3 (mod 127)

The above diagram provides all points in this group. It also illustrates an example of the reduced addition operation:

Given two points on the curve: P = (16,20) Q = (41,120) Draw a straight line passing through A and B, And wrap the line around when it reaches the boundary of the region. It will reach another point -R on the curve. Take the symmetric point R of -R: R = A + B R = (86,81)

Let's verify R = P+Q using algebraic equations given by the reduced elliptic curve group definition:

For any two given points on the curve: P = (x_{P}, y_{P}) = (16,20) Q = (x_{Q}, y_{Q}) = (41,120) R = P + Q is a third point on the curve: R = (x_{R}, y_{R}) Where: x_{R}= m^{2}- x_{P}- x_{Q}(mod p) (11) y_{R}= m(x_{P}- x_{R}) - y_{P}(mod p) (12) m(x_{P}- x_{Q}) = y_{P}- y_{Q}(mod p) (18) Calculation: m*(16-41) = 20-120 (mod 127) m*(-25) = -100 (mod 127) 102*m = 27 (mod 127) m = 27 * 1/102 (mod 127) m = 27*66 (mod 127) m = 1782 (mod 127) m = 4 x_{R}= 4*4 - 16 - 41 = -41 (mod 127) x_{R}= 86 y_{R}= 4*(16 - 113) - 20 = -300 (mod 127) y_{R}= 81 C = (86,81)

The result from algebraic equations matches the geometrical result!

Last update: 2019.

Table of Contents

Geometric Introduction to Elliptic Curves

Algebraic Introduction to Elliptic Curves

Abelian Group and Elliptic Curves

Discrete Logarithm Problem (DLP)

Generators and Cyclic Subgroups

►Reduced Elliptic Curve Groups

Converting Elliptic Curve Groups

Elliptic Curves in Integer Space

Python Program for Integer Elliptic Curves

Elliptic Curves Reduced by Modular Arithmetic

Python Program for Reduced Elliptic Curves

Point Pattern of Reduced Elliptic Curves

Integer Points of First Region as Element Set

Reduced Point Additive Operation

Modular Arithmetic Reduction on Rational Numbers

Reduced Point Additive Operation Improved

What Is Reduced Elliptic Curve Group

Reduced Elliptic Curve Group - E23(1,4)

Reduced Elliptic Curve Group - E97(-1,1)

►Reduced Elliptic Curve Group - E127(-1,3)

Reduced Elliptic Curve Group - E1931(443,1045)

Finite Elliptic Curve Group, Eq(a,b), q = p^n

tinyec - Python Library for ECC

ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

ECDSA (Elliptic Curve Digital Signature Algorithm)